The evolutionary mechanism of information negentropy spurred by emergencies

Abstract

In the context of social system resilience, the role of information negentropy—a measure of order and structural stability—remains underexplored during emergencies. This study investigates the evolutionary mechanisms of information negentropy under crisis conditions, aiming to quantify how emergencies disrupt and reshape social order through dual internal and external feedback. By integrating dissipative structure theory with a dynamic differential equation model, we propose a novel framework that captures the interplay between entropy increase and negentropy production. The model incorporates internal system dynamics (parameterized by $\:\alpha\:$) and external behavioral influences (parameterized by $\:\beta\:$), simulating how crises trigger transitions in social systems via “double-S” or “multi-S” evolutionary trajectories. Empirical validation using Baidu Index data from five brand-related public opinion crises demonstrates the model’s predictive accuracy, revealing that high-quality information dissemination and coordinated public behavior significantly amplify negentropy accumulation. Key findings indicate that emergencies elevate the upper limit of information negentropy by a factor of $\:{H}_{\infty\:}$, enabling systemic upgrades rather than mere disorder suppression. This study advances crisis management strategies by emphasizing proactive negentropy activation, offering actionable insights for enhancing social adaptability through real-time monitoring and parameter optimization. This work bridges theoretical gaps in entropy-driven social dynamics and provides a computational toolset for fostering resilience in complex information ecosystems.

Introduction

As an open system, the social system is essentially a typical dissipative structure¹. According to the principle of entropy increase, the ultimate fate of a closed system is the continuous increase of entropy until it reaches thermal death. In contrast, an open system can generate negentropy during the dissipation process through continuous exchanges of matter, energy, and information, thereby maintaining an orderly state. Traditional emergency management often focuses on “controlling negative impacts,” such as public opinion suppression and resource allocation, aiming to prevent the system from entering a high-entropy state of disorder. In this study, an “emergency” refers to a sudden and unexpected event that causes significant disruption or instability within the social system, typically characterized by high uncertainty, urgency, and widespread public attention. However, the true value of a crisis lies not only in controlling entropy increase but also in its potential to trigger negentropic flows, inducing system transitions and achieving a higher level of order. A crisis is not only a disruptive factor in the social system but also a crucial opportunity for negentropy accumulation.

The production, dissemination, and diffusion of information are key variables affecting social entropy changes, especially in an information-based society dominated by the internet, where information flow profoundly shapes the process of social entropy evolution. When a large amount of low-quality information emerges, efficient social energy is continuously converted into inefficient or ineffective forms², exacerbating the effect of increased entropy, increasing the uncertainty and disorder of the social system, and potentially pushing it to an entropy threshold under the stimulus of emergencies, leading to chaos. In contrast, if high-quality information is effectively transmitted, it can counteract the effect of increased entropy, reduce system uncertainty, and thus exert a negentropic effect in crisis situations³. As an essential component of the social system, the public opinion system’s information exchange mechanism directly determines the direction of entropy flow. If public institutions respond sluggishly, delay information disclosure⁴, or misguide public discourse during emergencies, public opinion entropy will further intensify, potentially causing the social system to fall into disorder. Conversely, if social actors can timely capture high-quality information, form rational cognition, and promote effective social coordination⁵, the system may accumulate negentropy and achieve a transition, making crises a catalyst for social evolution.

Therefore, the key to crisis management is not merely suppressing entropy increase but rather stimulating information negentropy to drive the social system toward a higher level of order. “Experience Brings Wisdom” (a phrase originally from the Chinese classic Zuo Qiuming’s Commentary on Spring and Autumn Annals, meaning that wisdom emerges from repeated engagement with crises and reflection on experience)—each emergency response process serves as a crucial phase for accumulating information negentropy and facilitating systemic transitions. This study is based on this core logic, constructing a corresponding analytical framework to reveal the intrinsic mechanisms of information negentropy evolution and exploring how to effectively harness negentropy effects in crisis scenarios. By doing so, it aims to enhance the adaptability and resilience of social systems and promote the progress of civilization.

Related work

The concept of entropy was first proposed by the German physicist K. Clausius to describe the irreversible flow of heat in a closed system, thereby revealing the principle of entropy increase, which states that an isolated system inevitably trends toward disorder in the absence of external exchanges. Later, physicist Erwin Schrödinger introduced the concept of negentropy in What Is Life?, arguing that living systems sustain order and counteract the disordering effects of entropy increase by extracting negentropy from their environment⁶. Since its introduction, negentropy theory has been widely applied not only in physics and biology but also in the field of social sciences. Trinn posited that social systems also adhere to entropy variation laws and comprise three subsystems—economic, cultural, and political—where negentropy plays a crucial role in maintaining order⁷. For instance, the economic system generates value through labor, the cultural system sustains stability through knowledge innovation, and the political system regulates social disorder through the operation of power. Carr-Chellman et al. further argued that the negentropy mechanism serves as a tool for the gradual development of social organizations, providing theoretical support for the sustainability of social systems⁸.

In recent years, negentropy theory has been widely applied across various fields and has gradually expanded toward the study of the dynamic evolution of social systems⁹. Existing research has explored the mechanisms of negentropy across multiple disciplinary contexts. In engineering and computer science, negentropy has been utilized to optimize physical algorithms, such as data compression and computational optimization based on spectral negentropy¹⁰. In urban management and social systems research, scholars have proposed rule-based negentropy models to enhance the efficiency of urban digital transformation¹¹. In environmental science, Wang et al. introduced a negentropy-based evaluation method to assess the sustainability of light particles in low-energy wastewater treatment¹². In industrial management, Durán et al. explored the potential of negentropy as a resilience measure based on Shannon entropy¹³. These studies demonstrate that the concept of negentropy is increasingly being applied across disciplines, showcasing extensive theoretical and practical value.

Despite the diversity of quantitative methods for negentropy in existing research, certain methodological limitations persist. Some studies overly rely on qualitative analysis and conceptual descriptions, lacking systematic quantitative analysis and empirical validation. Others have proposed specific quantitative methods but face limitations and assumptions in practical applications, particularly in non-engineering domains. For example, negentropy quantification methods based on Shannon’s entropy may be insufficient in capturing the complex interactions and information flows within social systems. In response to these gaps, this study integrates social entropy theory to construct an evolutionary framework for information negentropy under the stimulus of emergencies. It examines the intrinsic mechanisms of information negentropy evolution to explain how emergencies drive the enhancement of information negentropy and facilitate the self-optimization of social systems. Furthermore, by incorporating brand case studies, the study aims to assess the accuracy and validity of the proposed model, verifying the evolutionary patterns of negentropy theory in social systems. Ultimately, this research seeks to provide theoretical support for understanding how social systems achieve higher levels of stability through information adaptation in response to emergencies. These findings will further expand the application of entropy theory in social system analysis, offering new perspectives and analytical frameworks for studying information evolution in emergency contexts.

Modeling

Evolution mechanism of information negentropy

Sociologist and economist Kenneth Ewart Boulding pointed out that the most critical element in a social system is not mere data but knowledge. Brillouin further proposed that information itself constitutes negentropy, emphasizing that information can be transformed into negentropy and that negentropy can, in turn, be manifested through information¹⁴. Vaclav Reznicek and other scholars argued that information represents potential knowledge, while knowledge itself is negentropy¹⁵. Theoretically, the knowledge embedded within information enables its transformation into negentropy, highlighting that both knowledge and information serve as vital sources of negentropy. In social systems, the acquisition of negentropy relies not only on energy exchange but also on the acquisition and processing of information. The accumulation of information negentropy fundamentally involves the refinement and optimization of knowledge to construct a higher-order structure within the social system.

The evolution of information negentropy depends not only on the quantity of information but, more importantly, on its quality. During the process of information dissemination, the proliferation of low-quality information exacerbates entropy increase, heightening uncertainty within the social system. Conversely, the effective dissemination of high-quality information generates information negentropy, reducing disorder and enhancing systemic stability. In the context of emergencies, the evolutionary process of information negentropy manifests in the optimization and adjustment of individual and collective perspectives, attitudes, and cognition concerning specific events. This process not only influences the trajectory of entropy change within the system but also determines the degree of information negentropy evolution. As a crucial carrier of negentropy, the accumulation, dissemination, and reproduction of information dictate whether a social system can transition from disorder to a higher level of order when confronted with emergencies.

The concept of negentropy adopted in this study originates from the thermodynamic interpretation of entropy, which characterizes the degree of disorder and energy dispersal in an open system. Unlike Shannon entropy, which is defined in information theory as a measure of average uncertainty within a probabilistic distribution¹⁶, the notion of entropy here refers to the macroscopic structural degradation of order in social or informational systems exposed to external shocks. Negentropy, in this context, denotes the measurable reduction of systemic disorder resulting from the accumulation, transmission, and internalization of high-quality information. It reflects the degree to which information contributes to restoring or elevating systemic coherence under emergency conditions. This interpretation aligns with Schrödinger’s original conception of negentropy as the structural input required to sustain open systems away from thermodynamic equilibrium⁶. In this study, information negentropy’s evolution can be modeled mathematically through a differential framework that considers both endogenous system capacity and exogenous behavioral feedback. The resulting evolution equation is structurally derived from constrained growth dynamics and builds upon the foundational assumptions of self-limiting systems, as exemplified by the Logistic model.

In this context, let the information negentropy be denoted as $\:x$. In emergency scenarios, the accumulation of information negentropy is inherently limited. This is not only due to the finite structure of emergency-related information itself but also due to constraints in the public’s cognitive processing capacity and the bounded duration of the event lifecycle. Unlike theoretical entropy which may evolve indefinitely in open systems, the negentropy considered in this study emerges from a closed episode of collective cognition and behavioral alignment. Accordingly, we treat the upper bound $\:K$ of information negentropy as a structural ceiling determined by the system’s available information sources, attention span, and time horizon. When $\:\varDelta\:t\to\:0$, the instantaneous growth rate of $\:x$ over unit time, expressed as $\:\frac{\varDelta\:x}{x\varDelta\:t}$, is proportional to the remaining growth space $\:(1-\frac{x}{K})$¹⁷, with the proportionality constant denoted as $\:r$. Therefore, we can write:

$$\:\begin{array}{c}\frac{dx}{xdt}=r\left(1-\frac{x}{K}\right)\end{array}$$

(1)

The proportionality constant $\:r$ is also known as the maximum instantaneous growth rate per unit time. Rearranging Eq. (1), we get:

$$\:\begin{array}{c}\frac{dx}{dt}=rx\left(1-\frac{x}{K}\right)\end{array}$$

(2)

This is the Logistic model. The Logistic model is commonly used to describe biological growth¹⁷, population dynamics¹⁸, literature growth¹⁹, innovation diffusion²⁰, information propagation²¹, and other phenomena. In this paper, the Logistic model is applied to study the evolution of information negentropy. The mathematical essence expressed by the model is that the growth rate of information negentropy is directly proportional to the remaining growth space $\:(1-\frac{x}{K})$.

Quantifying the impact of emergencies on information negentropy

During the normal evolutionary process of information negentropy, the occurrence of emergencies often disrupts its existing trajectory, increasing uncertainty and instability in its evolution. The core characteristics of emergencies lie in their unpredictability and uncertainty. Therefore, quantifying the impact of emergencies on information negentropy requires a focus on how these events alter the evolutionary path of information negentropy and how they affect the stability of the information system.

The occurrence of emergencies typically impacts the social system, leading to an increase in social entropy and subjecting the information dissemination system to higher uncertainty. However, even under the complex and dynamic influence of emergencies, certain stable characteristics affecting information negentropy can still be identified. Upon analysis, we contend that, despite the significant uncertainty inherent in emergencies (i.e., unpredictability in time, location, and damage dynamics), their points of influence (i.e., observable behavioral and informational response channels such as search surges or discourse shifts) are relatively determinable. The observable influence points of emergencies—such as abrupt changes in search, consultation, or interaction behavior—do not merely alter the instantaneous state of information negentropy. They reshape the conditions under which further growth occurs by compressing or displacing the remaining potential for negentropy accumulation. Therefore, the primary effect of emergencies manifests as a disturbance to the growth potential of information negentropy. Specifically, within the framework of the Logistic model, the residual growth potential of information negentropy changes from $\:\left(1-\frac{x}{K}\right)$ to $\:\left(1-\frac{x}{K}+f\left(x\right)\right)$, where $\:f\left(x\right)$ is the emergency impact function. This function causes the future propagation pattern of information negentropy to deviate from the normal state, revealing a new evolutionary trend. Therefore, the evolutionary equation of information negentropy changes from Eq. (2) to

$$\:\begin{array}{c}\frac{dx}{dt}=rx\left(1-\frac{x}{K}+f\left(x\right)\right).\end{array}$$

(3)

For the Logistic model that describes the evolution of information negentropy, the Logistic model itself is essentially an autonomous information system. Therefore, the impact of emergencies on information negentropy can be regarded as the process of an emergency influencing the autonomous system. Hence, the impact of an emergency can be divided into two parts: the internal impact of the system, denoted as $\:{S}_{1}$, and the external impact of the system, denoted as $\:{S}_{2}$, represented by the equation

$$\:\begin{array}{c}f\left(x\right)={S}_{1}\left(x\right)+{S}_{2}\left(x\right).\end{array}$$

(4)

(1) Internal System Impact $\:{S}_{1}$: The emergency directly affects the internal dynamics of the information system, causing a change in the growth rate of information negentropy and pushing the system toward a new equilibrium. This impact is related to the system variable $\:x$. In this study, we assume internal system impact $\:{S}_{1}=\alpha\:\frac{x}{K}$, where $\:\alpha\:$ is an impact parameter and represents the relatively stable internal influence within the system.

(2) External System Impact $\:{S}_{2}$: The emergency indirectly affects the information system by inducing changes in the external environment, which in turn influences the evolution of information negentropy. Specifically, after an emergency occurs, the public’s behavior tends to adjust in response. Netizens (individuals who actively engage with information on digital platforms) may acquire information through online searches, browsing, and discussions, or may participate in offline activities or interventions related to the event. These external behaviors further impact the growth trend of information negentropy. In the context of an emergency, netizens’ information behavior typically follows three stages: “generation—diffusion—decline,” presenting an “S-shaped” evolutionary pattern. Therefore, we first assume that the emergency triggers a netizen behavior variable $\:{x}_{1}$ (representing the intensity or frequency of information-related behaviors among the public, such as online searching), whose variation follows the Logistic model, i.e.,

$$\:\begin{array}{c}\frac{d{x}_{1}}{dt}={r}_{1}{x}_{1}\left(1-\frac{{x}_{1}}{{K}_{1}}\right)\end{array}$$

(5)

Where $\:{r}_{1}$ represents the rate of change n the netizen behavior variable and $\:{K}_{1}$ represents the upper limit of the netizen behavior variable. Since the netizen behavior variable $\:{x}_{1}$ influences the information negentropy $\:x$ during its continuous growth, we assume that the external impact $\:{S}_{2}=\beta\:\frac{{x}_{1}}{{K}_{1}}\frac{x}{K}$, where $\:\beta\:$ is the external impact parameter, measuring the extent to which emergency-induced netizen behaviors affect the growth of information negentropy. The mechanism by which emergencies affect the evolution of information negentropy is illustrated in Fig. 1.

Fig. 1

Conceptual mechanism of emergency-driven information negentropy dynamics. The dashed line represents the evolution of information negentropy after the occurrence of the emergency.

Evolution model of information negentropy spurred by emergencies

Integrating the mechanisms of information negentropy evolution and quantifying the impact of emergencies, we derive a dual impact mechanism model of emergencies on information negentropy.

$$\:\begin{array}{c}\left\{\begin{array}{c}\frac{dx}{dt}=rx\left(1-\frac{x}{K}+\alpha\:\frac{x}{K}+\beta\:\frac{{x}_{1}}{{K}_{1}}\frac{x}{K}\right)\\\:\frac{d{x}_{1}}{dt}={r}_{1}{x}_{1}\left(1-\frac{{x}_{1}}{{K}_{1}}\right)\end{array}\right.\end{array}$$

(6)

Sorted out

$$\:\begin{array}{c}\left\{\begin{array}{c}\frac{dx}{dt}=rx\left(1-\left(1-\alpha\:-\beta\:\frac{{x}_{1}}{{K}_{1}}\right)\frac{x}{K}\right)\\\:\frac{d{x}_{1}}{dt}={r}_{1}{x}_{1}\left(1-\frac{{x}_{1}}{{K}_{1}}\right)\end{array}\right.\end{array}$$

(7)

The specific meanings of variables and parameters in the model are detailed in Table 1.

Table 1 Meanings of model variables and parameters.

Model analysis and simulation

Analysis of model equilibrium points and numerical solutions

Let

$$\:\begin{array}{c}\left\{\begin{array}{c}\frac{dx}{dt}=rx\left(1-\left(1-\alpha\:-\beta\:\frac{{x}_{1}}{{K}_{1}}\right)\frac{x}{K}\right)=0\\\:\frac{d{x}_{1}}{dt}={r}_{1}{x}_{1}\left(1-\frac{{x}_{1}}{{K}_{1}}\right)=0\end{array}\right.\end{array}$$

(8)

The equilibrium point of the model is

$$\:{P}_{1}\left(\text{0,0}\right),{P}_{2}\left(\frac{K}{1-\alpha\:},0\right),{P}_{3}\left(0,{K}_{1}\right),{P}_{4}\left(\frac{K}{1-\alpha\:-\beta\:},{K}_{1}\right)$$

Considering that the information negentropy information function$\:x\left(t\right)$ is a monotonically increasing function, only equilibrium point$\:{P}_{4}$ satisfies the model assumption. The stability conditions for the equilibrium point are $\:\alpha\:\ge\:0$, $\:\beta\:\ge\:0$, and $\:\alpha\:+\beta\:<1$. Let

$$\:\begin{array}{c}{H}_{\infty\:}=\frac{\frac{K}{1-\alpha\:-\beta\:}-K}{K}=\frac{\alpha\:+\beta\:}{1-\alpha\:-\beta\:}=-1+\frac{1}{1-(\alpha\:+\beta\:)}\end{array}$$

(9)

represent the degree of influence of emergencies on the increase in information negentropy. It is evident that $\:{\text{H}}_{{\infty\:}}$ is proportional to $\:\alpha\:+\beta\:$. Solving Eq. (6) yields $\:{x}_{1}\left(t\right)=\frac{{K}_{1}}{1+(\frac{{K}_{1}}{{x}_{1}\left(0\right)}-1)exp(-{r}_{1}t)}$. Therefore, the dual impact mechanism model transforms into:

$$\:\begin{array}{c}\frac{dx}{dt}=rx\left(1-\left(1-\alpha\:-\frac{\beta\:}{1+\left(\frac{{K}_{1}}{{x}_{1}\left(0\right)}-1\right){exp}\left(-{r}_{1}t\right)}\right)\frac{x}{K}\right).\end{array}$$

(10)

It can be seen that the essence of emergencies affecting information negentropy is to transform autonomous system $\:\frac{dx}{dt}=rx\left(1-\frac{x}{K}\right)$ into a non-autonomous system. This equation is of Bernoulli type, so the analytical solution of this equation is

$$\:\begin{array}{c}x\left(t\right)=\frac{x\left(0\right){exp}\left(rt\right)}{1+x\left(0\right){\int\:}_{0}^{t}\frac{rexp\left(rs\right)}{g\left(s\right)}ds}\end{array}$$

(11)

including $\:g\left(s\right)=K/(1-\alpha\:-\frac{\beta\:}{1+(\frac{{K}_{1}}{{x}_{1}\left(0\right)}-1)exp(-{r}_{1}s)})$.

Due to the integral term in expression $\:x\left(t\right)$, it is difficult to compute the zeros of higher-order derivatives. Therefore, to gain a better visual understanding of the model’s solution form and clarify the evolution pattern of information negentropy under the influence of emergencies, we arbitrarily select values within the parameter range to plot graphs of $\:x\left(t\right)$, $\:{x}_{1}\left(t\right)$, and their derivative functions, studying the numerical characteristics of the model solution. Let $\:r=0.5,\:K=1000,\:x\left(0\right)=0.01K,\:{r}_{1}=0.1,{\:K}_{1}=500,\:{x}_{1}\left(0\right)=0.01{K}_{1},\:\alpha\:=0.4,\:\beta\:=0.3$. The graphical representation of the model solution, along with its derivative function graph, are depicted in Fig. 2. It is straightforward to calculate that $\:x\left(t\right)$ approaches $\:\frac{\text{K}}{1-{\upalpha\:}-{\upbeta\:}}=3333.3333$ as $\:\text{t}\to\:+{\infty\:}$. The impact magnitude of emergencies on the increase of information negentropy is $\:{\text{H}}_{\infty\:}=2.3333$.

Fig. 2

Numerical simulation of the evolution of $\:x\left(t\right)$, $\:{x}_{1}\left(t\right)$ and their derivatives under parameter settings $\:r=0.5$, $\:K=1000$, $\:\:x\left(0\right)=0.01K$, $\:\:{r}_{1}=0.1$, $\:{\:K}_{1}=500$, $\:{x}_{1}\left(0\right)=0.01{K}_{1}$, $\:\alpha\:=0.4$, $\:\beta\:=0.3$.

• a. The trajectory of information negentropy $\:x\left(t\right)$ and the behavioral variable $\:{x}_{1}\left(t\right)$, both exhibiting S-shaped patterns. The “double-S” structure in $\:x\left(t\right)$ reflects the superimposed influence of internal system dynamics and external netizen behavior.

• b. The derivatives of $\:x\left(t\right)$ and $\:{x}_{1}\left(t\right)$, showing multiple inflection points and bimodal characteristics that correspond to distinct stages of negentropy evolution.

Observing Fig. 2-a, it is evident that in the absence of emergencies, $\:x\left(t\right)$ follows a Logistic model, characterized by a typical S-shaped curve. Following the occurrence of emergencies, when only internal system influence $\:{S}_{1}$ exists, $\:\beta\:=0$, at this time $\:\frac{dx}{dt}=rx\left(1-(1-\alpha\:)\frac{x}{K}\right)$, still exhibits a typical S-shaped curve structure. When both internal and external system influences are present, the behavioral variable $\:{x}_{1}\left(t\right)$ of netizens conforms to the Logistic model’s S-shaped curve structure. At this point, $\:x\left(t\right)$ exhibits a “double-S” structure, where the first S structure is mainly caused by internal system influence, and the second S structure is primarily induced by external system influence. $\:x\left(t\right)$ can be understood as the “superposition” of internal and external influences, which forms the paradigm constructed in this paper. Observing the derivative function graph (Fig. 2-b), it is found that the inflection point of the information negentropy evolution curve before the impact of emergencies is at $\:\left(\text{9.1902,500}\right)$, while after the influence of emergencies, the inflection points are at $\:\left(\text{10.2487,844.1763}\right)$, $\:\left(\text{23.3849,1733.6867}\right)$, and $\:\left(\text{54.9717,2501.3779}\right)$. The number of inflection points increases from one before the emergency to three after it, with the function increment between the first and third inflection points being the largest, reaching $\:\frac{2501.3779-844.1763}{3333.3333}=49.7160\text{\%}$. The derivative variable of information negentropy evolution corresponds to the statistical data of information negentropy evolution, leading to a bimodal phenomenon in the statistical data, which corresponds to the “double S” structure.

Dynamic simulation under the stimulation of emergencies

To gain an intuitive understanding of the dynamic impact of emergencies, we fix the information negentropy and netizen behavior variables under non-emergency conditions. Within the range of parameter variations, we arbitrarily select values to plot the changes in variable $\:x\left(t\right)$ after different time points of emergency intervention.

For the information negentropy evolution model in the absence of emergencies (Eq. (2)), setting parameters as $\:r=0.5,K=\text{1000,0}\le\:t\le\:70$, with an initial value of $\:0.01K$, the third derivative of $\:x\left(t\right)$ is computed (the third derivative is often used in lifecycle modeling to determine acceleration extrema, which correspond to the transitions between latent, diffusion, and decline stages¹⁷, yielding zero points at $\:{t}_{1}=7.0066$ and $\:{t}_{2}=12.2745$. For the evolution model of the online behavior variable induced by emergencies (Eq. (5)), setting parameters as $\:{r}_{1}=0.2$, $\:{K}_{1}=500$, with an initial value of $\:{0.01K}_{1}$. ① For Eq. (2), the numerical solution graph of information negentropy evolution under non-emergency conditions was computed. ② At $\:t=0:2:20$, the influence of emergencies was introduced, and for Eq. (6), the parameters $\:\alpha\:=0.1$ and $\:\beta\:=0.2$ were applied. Based on these settings, the corresponding numerical solution graph of the model was obtained (Fig. 3).

Fig. 3

Numerical solution of $\:x\left(t\right)$ and $\:{x}_{1}\left(t\right)$ under emergency intervention. Simulated with parameters $\:r=0.5$, $\:K=1000$, $\:\alpha\:=0.1$, $\:\beta\:=0.2$, $\:{r}_{1}=0.2$, $\:{K}_{1}=500$. The curve of $\:x\left(t\right)$ transitions from a “double-S” to a “triple-S” structure as emergency inputs are introduced at different times, while $\:{x}_{1}\left(t\right)$ reflects the corresponding behavioral response with a noticeable time lag. The two vertical dashed lines denote the zero points of the third derivative of $\:x\left(t\right)({t}_{1}=7.0066,\:{t}_{2}=12.2745)$, which divide the process into three stages: latent, diffusion, and decline.

Observations indicate that during the dynamic influence of emergencies on information negentropy, the curve $\:x\left(t\right)$ in the interval $\:\left[\text{0,70}\right]$ transitions from a “double-S” structure to a “triple-S” structure. Although the dynamic “superposition” pattern of emergency impacts varies, $\:x\left(t\right)$ ultimately converges to the equilibrium point $\:\frac{K}{1-\alpha\:-\beta\:}=1428.57$. This suggests that while the dynamic influence process differs, the overall trend remains consistent. Additionally, the three inflection points of $\:x\left(t\right)$ shift to the right, with the third inflection point aligning with the corresponding inflection point of $\:{x}_{1}\left(t\right)$, though with a noticeable time lag. This indicates that the “superposition” of emergency impacts entails a certain reaction time. Furthermore, in the interval $\:[0,{t}_{1}]$, the influence of emergencies on information negentropy is minimal, closely resembling the evolution curve of information negentropy in the absence of emergencies, conforming to the Logistic curve pattern. The available data for model parameter estimation in this phase is relatively sparse, making it less suitable for monitoring emergency impacts. In the interval $\:[{t}_{1},{t}_{2}]$, before $\:\frac{{t}_{1}+{t}_{2}}{2}$, the curve still approximates a Logistic function, whereas after $\:\frac{{t}_{1}+{t}_{2}}{2}$, the impact of emergencies on information negentropy gradually increases. In the interval $\:[{t}_{2},+{\infty\:}]$, the influence of emergencies on information negentropy becomes pronounced, with a rapid initial increase in information volume. This phase provides abundant data for model parameter estimation, making it valuable for real-time monitoring of emergency impacts and predicting the evolutionary trend of information negentropy under emergency influences.

Reverse simulation for model parameters

To achieve a quicker recovery of the social system after emergencies, we need to adjust the values of $\:\alpha\:$ and $\:\beta\:$ through reverse simulation methods. This will allow us to study the correlation between the recovery speed of the social system after the impact of emergencies and the model parameters $\:\alpha\:$ and $\:\beta\:$. The goal is to find parameter combinations that enable $\:t=0$ to return to a steady state earliest at time $\:x\left(t\right)$. The core of reverse simulation lies in deducing key parameters not from the system’s initial state but from its known evolutionary outcomes. In this study, this means starting from the steady state of $\:x\left(t\right)$ and working backwards to solve for parameters $\:\alpha\:$ and $\:\beta\:$.

We fix the information negentropy evolution in the absence of emergencies and, within the parameter variation range, select arbitrary values to plot the graph of $\:x\left(t\right)$ f under different values of the emergency impact parameters $\:\alpha\:$ and $\:\beta\:$ (Fig. 4). For the information negentropy evolution model without emergencies (Eq. (2)), we set the parameters as $\:r=0.5,\:K=\text{1000,0}\le\:t\le\:120$, with an initial value of $\:0.01K$. For the emergency-induced public behavior variable evolution model (Eq. (5)), we set the parameters as $\:{r}_{1}=0.2$, $\:{K}_{1}=500$, with an initial value of $\:{0.01K}_{1}$. ① For Eq. (2), we plot the numerical solution of information negentropy evolution in the absence of emergencies. ② We set $\:\alpha\:=0.1:0.1:0.4$ and $\:\beta\:=0.1:0.1:0.4$, introduce the impact of emergencies, and plot the inverse simulation graph of the model based on Eq. (6).

Fig. 4

Inverse Simulation Graph of the Model.

Upon observation, it is found that reducing parameter $\:{\upalpha\:}$ alone causes the second and third inflection points of $\:x\left(t\right)$ to noticeably shift downward. Specifically, decreasing the internal system impact alone reduces the negative influence on information negentropy within the system, leading to a slower growth rate in the mid-term, reflecting a slowdown in the rate of entropy increase within the system. Additionally, before the system reaches a new stable state, the growth of negentropy information is significantly more inhibited. Furthermore, reducing parameter $\:\beta\:$ alone causes the second inflection point of $\:x\left(t\right)$ to slightly shift downward to the right, with a clear trend of the third inflection point shifting downward to the left. This shift indicates a reduction in external negative impacts on the system, causing a slowdown in the initial growth rate of information negentropy. As the driving force from outside the system weakens, the speed at which the system reaches a stable state increases, and the eventual stable state may be achieved at a lower level of negentropy. With an increase in parameter $\:\alpha\:+\beta\:$, the “double S” feature of the numerical solution plot becomes progressively more pronounced, clearly showing that the function increment is maximal between the first and third inflection points of $\:x\left(t\right)$. Additionally, when only parameter $\:\beta\:$ is increased, the growth rate of $\:x\left(t\right)$ from the first to the second inflection point is significantly lower compared to when only parameter $\:{\upalpha\:}$ is increased. However, the growth rate from the second to the third inflection point is noticeably higher than when only parameter $\:\alpha\:$ is increased. This indicates that under the influence of emergencies, before the first and second inflection points, the impact of internal parameter $\:\alpha\:$ on information negentropy is greater than that of external parameter $\:\beta\:$. In other words, the function increment within the same time period is larger. Before the second and third inflection points, the impact of external parameter $\:\beta\:$ on information negentropy is greater than that of internal parameter $\:\alpha\:$. The main reason is that in scenario $\:\alpha\:\frac{x}{K}$, parameter $\:{\upalpha\:}$ directly plays a role, while in scenario $\:\beta\:\frac{{x}_{1}}{{K}_{1}}\frac{x}{K}$, parameters $\:\beta\:$ and $\:\frac{{x}_{1}}{{K}_{1}}$ act together. The growth of $\:\frac{{x}_{1}}{{K}_{1}}$ takes some time but will have a greater impact later on.

Through reverse simulation, we can not only determine the parameter values that enable rapid system recovery but also gain a deeper understanding of how parameter changes affect the evolution of information negentropy. This provides theoretical foundations and data support for managing and making decisions about the impacts of emergencies in practical applications. Furthermore, the reverse simulation process demonstrates the flexibility and adaptability of the model, allowing adjustments to model behavior based on various social needs and objectives to achieve anticipated societal benefits.

Estimation methods for model parameters

Before the emergency occurs, the evolution model of information negentropy is Eq. (3). It transforms into its corresponding difference equation

$$\:\begin{array}{c}\varDelta\:x\left(k\right)=rx\left(1-\frac{x}{K}\right)=rx-\frac{r}{K}{x}^{2}.\end{array}$$

(12)

In which $\:\varDelta\:x\left(k\right)=x\left(k\right)-x(k-1)$, $\:k=\text{1,2},3...$. Upon observation, it is found that there exists a bivariate linear function relationship between $\:\varDelta\:x\left(k\right)$ and $\:x$, $\:{x}^{2}$. Specifically, $\:\varDelta\:x\left(k\right)$ is the dependent variable, and $\:x$, $\:{x}^{2}$ are the independent variables in a multiple linear regression. Therefore, under the premise of known information data before the impact of emergencies, regression coefficients $\:r$ and $\:-\frac{r}{K}$ can be obtained through multiple regression analysis, thereby obtaining model parameters $\:r$ and $\:K$.

To ensure the reproducibility and validity of the estimation, we first observe that in practical scenarios, both the intrinsic growth rate $\:r$ and the upper bound $\:K$ of information negentropy should be strictly positive. Therefore, structural testing criteria $\:-\frac{r}{K}<0$ can be established based on these conditions.

Furthermore, to ensure more accurate and representative results, we have also introduced goodness-of-fit and significance tests. The goodness-of-fit test introduces the statistical metric R-squared, which measures the fit of a linear regression model. Typically, a higher R-squared indicates better fit of the regression model to the observed data. We set a criterion to pass the goodness-of-fit test when R-squared exceeds 0.36. The significance test introduces a threshold value denoted as $\:P-value$, which is based on actual statistical metrics to determine the significance level. We set a criterion to pass the significance test when the value $\:P-value<0.05$ is satisfied²².

After the occurrence of emergencies, the evolution model of information negentropy (Eq. (6)) transforms into its corresponding system of difference equations

$$\:\begin{array}{c}\left\{\begin{array}{c}\varDelta\:x\left(k\right)-rx+\frac{r}{K}{x}^{2}=rx\left(1-\left(1-\alpha\:-\beta\:\frac{{x}_{1}}{{K}_{1}}\right)\frac{x}{K}\right)=\frac{r\alpha\:}{K}{x}^{2}+\frac{r\beta\:}{{KK}_{1}}{{x}_{1}x}^{2}\\\:\varDelta\:{x}_{1}\left(k\right)={r}_{1}{x}_{1}\left(1-\frac{{x}_{1}}{{K}_{1}}\right)={r}_{1}{x}_{1}-\frac{{r}_{1}}{{K}_{1}}{x}_{1}^{2}\end{array}\right.\end{array}$$

(13)

In which $\varDelta\:x\left(k\right)=x\left(k\right)-x(k-1),\varDelta\:{x}_{1}\left(k\right)={x}_{1}\left(k\right)-{x}_{1}(k-1)$, $\:k=\text{1,2},3,...$. Upon observation, it is found that there exists a bivariate linear function relationship between $\:\varDelta\:x\left(k\right)-rx+\frac{r}{K}{x}^{2}$ and $\:{x}^{2}$, $\:{{x}_{1}x}^{2}$, and between $\:\varDelta\:{x}_{1}\left(k\right)$ and $\:{x}_{1}$, $\:{x}_{1}^{2}$. Therefore, under the premise of known initial information quantity impacted by emergencies and data variables of netizen behavior, through multiple regression analysis, regression coefficients $\:\frac{r\alpha\:}{K}$, $\:\frac{r\beta\:}{{KK}_{1}}$, $\:{r}_{1}$, $\:-\frac{{r}_{1}}{{K}_{1}}$ can be obtained, thereby deriving model parameters $\:\alpha\:$, $\:\beta\:$, $\:{r}_{1}$ $\:{K}_{1}$. Model parameters $\:r$, $\:K$, $\:\alpha\:$, $\:\beta\:$, $\:{r}_{1}$ $\:{K}_{1}$ can be obtained through simple calculations. Furthermore, after determining the model parameters using this method, based on initial values $\:x\left(0\right)$, $\:{x}_{1}\left(0\right)$, model solution curves can be plotted according to the model. This allows for the prediction of the evolution trends of information negentropy under the impact of emergencies.

Empirical analysis

Data source

This study selects five representative brand public opinion emergencies on Chinese social media over the past year as empirical samples (Table 2). The sample selection follows these principles: (1) Baseline of normal information negentropy: The chosen brands must exhibit stable public discussion in the absence of emergencies, reflecting the baseline level of information flow in the social system under normal conditions. (2) Significant emergency-induced disturbance: The event must lead to a surge in public opinion intensity within a short period (typically 24–72 h), aligning with the dissipative structure theory’s characteristic of “critical fluctuations triggering system phase transitions.” (3) Data traceability: The event’s dissemination cycle must be complete, with key information sources traceable to explicit nodes in media reports and public interactions.

Table 2 Brand public opinion Emergency.

The empirical data are sourced from the Baidu Index (https://index.baidu.com/v2/index.html#/). The Baidu News Index quantifies the intensity of public information dissemination and the trajectory of emotional evolution by aggregating the frequency of news reports on specific keywords and weighting users’ reading, commenting, and sharing behaviors. Meanwhile, the Baidu Search Index calculates the weighted frequency of keyword searches on the Baidu platform, capturing the dynamics of public attention and information demand regarding an event.

The disorder within a social system (entropy increase) originates from information chaos and cognitive conflicts, whereas the accumulation of negentropy is realized through the proactive dissemination of high-quality information by the public, such as fact clarification and rational discussions. The Baidu News Index, by tracking the spatiotemporal evolution of user interactions, can indirectly measure the initial controversy of an emergency (entropy disturbance) and the rapid diffusion of readership and positive comments driven by public engagement (negentropy generation). Simultaneously, the Baidu Search Index reflects proactive search behaviors, indicating the sustained external input of group participation into system evolution. According to dissipative structure theory, such behavioral interactions provide external energy to counteract entropy increase, facilitating the cross-network diffusion of negentropy and reconstructing system stability.

Data modeling

Modeling the evolution of information negentropy in the absence of emergencies

Selecting a natural evolution cycle before each brand’s emergency event as the baseline for modeling information negentropy in the absence of emergencies. The specific timeframe for selecting baseline information negentropy data is shown in the Table 3.

Table 3 Time period of normal information negentropy data Quantity.

Transform Eq. (2) into corresponding difference equations $\:\varDelta\:x\left(k\right)=rx\left(1-\frac{x}{K}\right)=rx-\frac{r}{K}{x}^{2}$, where $\:\varDelta\:x\left(k\right)=x\left(k\right)-x(k-1)$ and $\:k=\text{1,2},3,...$. Then, a linear regression analysis is conducted based on the baseline data of information negentropy. The results are subsequently evaluated using the three validation criteria introduced in Sect. 4.4. Finally, determining the coefficients and presenting the fitted results in the Table 4.

Table 4 Results of static regression Fitting.

Modeling the evolution of information negentropy under the stimulation of emergencies

Dynamic data modeling of information negentropy data and variables of netizen behaviors. Dynamic regression analysis is conducted for each brand’s public opinion emergency event starting from the point where the data volume satisfies the triple validation criteria. (Dynamic regression analysis refers to a process where regression is performed incrementally—beginning with a partial data set, adding one data point at a time, and continuing until the full-period data is covered.)

First, based on the first equation in Eq. (13), the regression coefficients $\:\frac{r\alpha\:}{K}$ and $\:\frac{r\beta\:}{{KK}_{1}}$ are calculated. Observing the $\:P-value$ and the coefficient of determination $\:{R}^{2}$, it is found that the regression analysis fitting effects are all highly significant. Partial results of dynamic regression fitting are shown in the Table 5.

Table 5 Results of dynamic regression fitting for information Negentropy.

Secondly, based on the second equation in Eq. (13), regression analysis is conducted on the behavioral variable data of netizens corresponding to the dynamic data volumes. The regression coefficients $\:{r}_{1}$ and $\:-\frac{{r}_{1}}{{K}_{1}}$ are calculated, and by observing the $\:P-value$ and the coefficient of determination $\:{R}^{2}$, it is found that the results of the regression analysis are highly significant, as shown in Table 6.

Table 6 Dynamic regression fitting results of netizens’ behavioral Variables.

In summary, the regression analysis results for both models across the five case studies demonstrate statistically significant fitting effectiveness. Based on these results, we compute the model parameters $\:\alpha\:$, $\:\beta\:$ and equilibrium point $\:\frac{K}{1-\alpha\:-\beta\:}$(Table 7). In addition, following the completion of dynamic modeling, the model was further employed for short-term forecasting of information negentropy. For instance, in the case of Florasis, parameters estimated at $\:n\:=\:20$ were used to predict subsequent values. Across all five cases, the predicted trajectories closely matched the observed data (Fig. 5). Evaluation using Mean Absolute Percentage Error (MAPE) indicates that the prediction errors remain relatively low during the mid-to-late stages of each event. These results provide empirical support for the validity of the dual-impact mechanism model constructed in this study and further demonstrate the model’s capability to forecast the influence trajectory of emergencies on information negentropy.

Table 7 Evaluation of the predictive ability of the Model.

Fig. 5

Comparison of Fitted and Actual Values of Information Negentropy Across Five Brand Public Emergencies.

Discussion

Comparative analysis

In this study, we utilized Baidu Index as a primary data source to measure one of the indicators of information negentropy. In comparison, Google Trends represents another commonly used data source for analyzing group behaviors and event impacts. A pioneering study in this area is Choi and Varian²³, which demonstrated that Google Trends data can be employed for short-term forecasting of economic activities including retail, automotive sales, housing sales, and tourism. A review by Nuti et al. on Google Trends indicated that among 70 studies conducted from 2009 to 2013²⁴, 70% utilized time trend analysis (comparisons across time periods), 11% employed cross-sectional analysis (comparisons across different locations within a single time period), and 19% utilized both methods simultaneously. Furthermore, researchers have employed various statistical analyses in conjunction with Google Trends data, including correlation analysis, analysis of variance, t-tests, multiple linear regression, continuous density hidden Markov models, Box-Jenkins transfer function models, time series analysis, and Mann-Whitney tests. As a novel statistical analysis platform, Baidu Index records the web browsing data of hundreds of millions of Chinese internet users, playing a crucial role in the era of big data. Based on web and news searches, Baidu Index reflects the popularity of keywords on Baidu’s search platform, including search frequency, demographic characteristics of searchers, and the popularity of related news, which provides valuable insights for businesses in making marketing decisions. Baidu Index offers advantages in terms of regional adaptability and data update frequency, primarily targeting the Chinese market with daily updates that enable real-time monitoring of changes in group behavior. In contrast, Google Trends covers global data but has longer update cycles. Past studies on Google Trends have primarily focused on methods such as correlation analysis, predictive models, spatial analysis, and studies on influencing factors, but have not extensively compared with Baidu Index. Future research could explore the combined use of different data sources and delve deeper into their respective strengths, weaknesses, and applicability to obtain more comprehensive and accurate research results. Both Google Trends and Baidu Index are widely used tools for web search data analysis, having been applied across diverse fields including economics²⁵, politics²⁶, disease trends²⁶, and mental health²⁷. Compared to traditional survey and measurement methods, web search data can provide more timely and rich results, often correlating well with outcomes obtained through opinion polls or traditional measurement methods²⁸.

Compared to previous studies on negentropy, our research demonstrates significant advancements and advantages. Existing literature predominantly relies on static indicators, such as total information volume and entropy thresholds, to measure the orderliness of social systems. However, these approaches overlook the dynamic feedback mechanisms of emergencies, making it difficult to explain the process of “crisis-triggered negentropy activation and system reconstruction.” By constructing a differential equation model, this study enables a quantitative analysis of the evolutionary mechanism of information negentropy, thereby improving analytical precision and enhancing the objectivity of results. The proposed model possesses strong predictive capabilities, allowing for the forecasting of how emergencies impact information negentropy, thus providing decision-makers with valuable forward-looking insights. Furthermore, this study delves into the evolutionary process of information negentropy and empirically validates the model’s rationality, ensuring the reliability of the research findings. A negentropy control method based on model parameters is proposed, offering practical guidance for mitigating the adverse effects of emergencies on social systems. Additionally, the model accounts for the unique characteristics of social systems, such as the complexity of collective behavior, making it more applicable to real-world scenarios. By integrating both data-driven and theory-driven approaches, this study establishes a more comprehensive and in-depth analytical framework for future research. These advantages not only enrich the theoretical understanding of negentropy but also provide new strategies and tools for the management and response to emergencies in practice.

Applications and directions

(1) Integrated analysis and prediction. Integrated analysis and prediction are two critical components in studying social systems. The alignment error between analytical and predictive models directly affects the accuracy of forecasting social system evolution trends. By collecting and analyzing real-time data from social systems such as online public opinion, media reports, and public behaviors, we can monitor the current state of information negentropy and identify any anomalies or trend changes. Combining real-time data allows us to predict the evolution trends of information negentropy under specific impacts of emergencies. The parameters $\:{\upalpha\:}$ and $\:{\upbeta\:}$ in the model can be adjusted based on actual conditions to reflect real-time changes in the influences both within and outside the system. Based on the model presented in this paper, an analysis and prediction of the evolution trends of information negentropy can be conducted, addressing the integrated issue of “analysis-prediction.” In particular, constructing a continuous model that evolves over time will enhance the interpretability of the correlations, endowing them with a philosophical foundation related to social evolutionary mechanisms. This approach ultimately aims to achieve a combination of data-driven and theory-driven methodologies.

(2) Crisis-Driven Negentropy Transition. Traditional crisis management primarily focuses on “mitigating negative impacts” through measures such as public opinion suppression and resource allocation. However, the model proposed in this study reveals that the true value of a crisis lies in its ability to trigger a chain reaction of negentropy excitation, accumulation, and transition, ultimately facilitating an upgrade in systemic order. The conclusions drawn from the simulation provide quantitative guidance for social systems in adapting to crises. By adjusting the internal influence parameter $\:\alpha\:$ and the external influence parameter $\:\beta\:$, the evolutionary trajectory of information negentropy can be dynamically optimized, enabling a shift from passive response to proactive adaptation. Specifically, $\:\alpha\:$ represents the system’s internal efficiency in processing information, such as government response speed and information transparency. Enhancing $\:\alpha\:$ directly strengthens the system’s intrinsic capacity for negentropy generation. In contrast, $\:\beta\:$ reflects the synergistic effects of external collective behavior, such as rational public discussions and the dissemination of high-quality information. Increasing $\:\beta\:$ accelerates the cross-network diffusion of negentropy. Numerical simulations indicate that when $\:\alpha\:+\beta\:<1$, the equilibrium point $\:\frac{K}{1-\alpha\:-\beta\:}$ rises significantly with increasing parameters, leading to a synchronous increase in the impact degree $\:{H}_{\infty\:}$. This implies that, in the early stages of a crisis, rapidly increasing $\:\beta\:$ —for instance, by establishing open information-sharing channels and encouraging public participation in fact verification—can swiftly suppress entropy disturbances and reduce the time required for the behavioral variable $\:{x}_{1}\left(t\right)$ to reach saturation. In the mid-to-late stages of a crisis, strengthening $\:\alpha\:$ through measures such as optimizing public opinion monitoring algorithms and establishing cross-departmental collaboration mechanisms can consolidate the stability of internal negentropy accumulation, driving the system toward a higher level of order. Furthermore, inverse parameter simulations reveal a threshold effect in the coordinated regulation of $\:\alpha\:$ and $\:\beta\:$. When $\:\alpha\:$ dominates the early-stage evolution, the system’s internal negentropy generation efficiency is maximized. Conversely, when $\:\beta\:$ takes the lead in later stages, external synergy significantly amplifies the scope of negentropy diffusion. This dynamic adaptation mechanism provides policymakers with a stratified intervention strategy: in the temporal dimension, resource priorities can be flexibly allocated based on different crisis phases, while in the spatial dimension, parameter combinations can be adjusted to account for variations in community characteristics. For instance, for information-sensitive groups, enhancing $\:\alpha\:$ through algorithmic optimization can facilitate precise dissemination of authoritative information, whereas for highly participatory groups, increasing $\:\beta\:$ through incentive mechanisms can guide them into becoming key nodes for negentropy propagation. Ultimately, the model operationalizes the concept of crisis adaptation by transforming it into a quantifiable and predictable negentropy regulation process. This ensures that the wisdom encapsulated in the adage Experience Brings Wisdom can be systematically applied within complex social systems, making it an implementable engineering solution.

Although the model proposed in this study provides a framework for understanding the evolutionary process of information negentropy and the impact of emergencies on it, there are several limitations that need to be considered. First, the model assumes that the evolution of information negentropy is influenced by the internal system influence coefficient $\:\alpha\:$ and the external system influence coefficient $\:\beta\:$. In the real world, group behavior is affected by various factors such as culture, social environment, and political elements. The simplifying assumptions of the model may limit its ability to accurately describe the complexity of real-world group behavior. Furthermore, when considering the impact of emergencies on information negentropy, the model primarily focuses on the influence of emergencies on netizen behavior, while the impact of other factors, such as the potential psychological effects of emergencies on society, may not have been fully considered. Therefore, this study provides an initial framework for understanding the evolution of information negentropy and the influence of emergencies, but further research and exploration are needed to refine the model in order to better address complex real-world issues.

Additionally, the definition of information negentropy adopted in this study is particularly suited to modeling the accumulation and transformation of high-quality information within open social systems, such as public opinion events and digital behavioral responses. It may not be directly applicable to other domains—such as physical or biological systems—where the meaning of “order” is not primarily derived from information flow. Misinterpretations could arise if the proposed formulation is applied outside the theoretical context of information-based systemic dynamics.

Finally, while entropy-related processes are often fundamentally stochastic in nature, the present model adopts a deterministic approach to describe the average macroscopic trajectory of negentropy evolution. This choice is consistent with common practices in social systems modeling, where individual behaviors may be random but aggregate patterns are structurally constrained and analyzable. Nonetheless, the incorporation of stochastic components—such as uncertainty in public attention or variability in information dissemination—would offer a more comprehensive picture of negentropy dynamics. We acknowledge this as a valuable direction for future research.

Conclusion

This study systematically reveals the dynamic feedback mechanism of social systems under the stimulation of emergencies by constructing a differential equation model of information negentropy evolution. The results indicate that the essential value of emergencies lies not only in their disturbance effects on social systems but also in their intrinsic logic that triggers system transitions through the excitation and accumulation of information negentropy. The extended framework based on the Logistic model shows that emergencies reconstruct the information negentropy evolution path through a dual influence mechanism from both within and outside the system: the internal influence parameter $\:\alpha\:$ directly affects the inherent growth pattern of information negentropy, while the external influence parameter $\:\beta\:$ forms a dynamic superimposition effect through the “S-shaped” diffusion of netizen behavior. Together, these mechanisms drive information negentropy to break through the normal evolution trajectory, presenting complex structures such as “double S” or even “multiple S”. Analysis of the model’s equilibrium points and numerical simulations confirm the core proposition of “crisis-driven negentropy transition” — when $\:\alpha\:+\beta\:<1$, emergencies can lead to an increase in the upper limit of information negentropy by a factor of $\:{H}_{\infty\:}$, providing mathematical support for the evolutionary law of “Experience Brings Wisdom”.

Empirical research on five types of brand crisis events shows that the predicted model curve aligns closely with the real data evolution trend, confirming that the accumulation of information negentropy is essentially a dynamic process of public cognitive optimization and collective behavior coordination. This research breaks through the limitations of traditional crisis management, which focuses on controlling entropy increase, and proves that timely guidance of high-quality information dissemination and the stimulation of positive behavior feedback from netizens can transform a crisis into a catalyst for the orderly upgrading of the social system. This integrated analysis-prediction framework not only expands the application boundaries of dissipative structure theory in social system research but also reveals the key pathways for negentropy regulation through inverse simulation of parameters. It provides decision-makers with actionable theoretical tools to achieve the virtuous cycle of “crisis learning - system evolution.” The study ultimately elucidates that, as a special driving force of information negentropy evolution, the deep value of emergencies lies in their ability to promote a spiral upward accumulation of wisdom in social systems through cognitive reconstruction. This is the scientific interpretation of the “Experience Brings Wisdom” proposition in complex social systems.